Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:24:03Z http://mathoverflow.net/feeds/question/51085 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51085/subfields-of-mathbbc-isomorphic-to-mathbbr-that-have-baire-property Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property Ricky Demer 2011-01-04T03:36:46Z 2011-01-06T04:10:12Z <p>While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then wondered whether this was unique. When I got back to my computer, I looked on google and came up with <a href="http://camoo.freeshell.org/cfield.isor" rel="nofollow">this</a>, which indicated that is not. However, the proof given there makes heavy use of Choice, and based on similar things, I am guessing that this cannot be avoided. <br><br><br> Define DC($\omega_1$) as: <br><br> For all <a href="http://en.wikipedia.org/wiki/Tree_%28set_theory%29" rel="nofollow">trees</a> $T$, $\quad$ $T\:$ has a branch $\$ or $\$ $T\:$ has a chain of length $\omega_1$ $\quad$. <br><br> which, if I haven't messed up the simplification, ZF proves is equivalent to the definition given at shelah.logic.at/files/446.ps . <br><br><br><br> Does $\ ZF+DC(\omega_1)\$ prove that there is a unique subfield of $\mathbb{C}$ which is both isomorphic to $\mathbb{R}$ and has Baire property?</p>